I think certain pets could be strategic survival allies. Dogs for intruder detection, cats if you are in an agrarian situation and need rodent control to protect the crops.
The problem will be people who can only keep pets when the economy is relatively stable. Already pet abandonment is a serious problem. The problem pets will be the big, already somewhat vicious dogs, that some folks keep. In my neighborhood there is already a very serious problem with hazard dogs on the loose, and their owners tend to be people who will be among the first in trouble. There is already a problem with pet-abandonment, due to unemployment.
Dogs make good alarms but poor defense. A family with two big dogs was robbed shortly after we moved here. It was really easy: the burglars simply took chairs, backed the dogs into another room, and closed the door on them. Dogs can't open doors.

I've heard of other crooks baiting them into similar situations with meat. The crooks cleaned the family out, and worked at their leisure. The rest of us totally ignored the alarm that went off, because there had already been too many false alarms.